# Sequences

• A sequence is an ordered list of values.
• e.g. the sequence of powers of two: $$1,2,4,8,16,32,64,\ldots$$.
• e.g. the sequence of digits: $$0,1,2,3,4,5,6,7,8,9$$.
• We can think of a sequence as a function mapping the natural numbers ($$\{0,1,2,3,\ldots\}$$ or some subset) to the values in the sequence.
• e.g. $$f(n)=2^{n-1}$$
• Usually when thinking of it as a sequence, we will label the elements $$a_n$$, so would write $$a_n=2^{n-1}$$ instead.
• We can start the sequence with either $$n=0$$ or $$n=1$$.
• Whichever is more convenient.
• Or could start anywhere else if we needed to.
• So, would probably have written the above as $$a_n=2^n$$.
• An arithmetic progression is a sequence where each term differs by a real number: $a,a+d,a+2d,a+3d,\ldots\,.$
• So, term $$n$$ is $$a_n=a+dn$$.
• An geometric progression is a sequence where each term is multiplied by a constant factor: $a,ar,ar^2,ar^3,\ldots\,.$
• So, term $$n$$ is $$a_n=ar^n$$.
• The powers of two above were a geometric progression with $$a=1$$ and $$r=2$$.

# Summations

• A summation is what you get when you add up a sequence.
• For a sequence with terms up to $$a_n$$, its summation is $$a_0+a_1+\cdots +a_n$$.
• This is usually written (for finite and infinite sequences) $\sum_{i=0}^{n} a_i \qquad \sum_{i=0}^{\infty} a_i\,.$
• Theorem: For real numbers $$a$$ and $$r$$, with $$r$$ not 0 or 1, $\sum_{i=0}^{n} ar^i = a\frac{r^{n+1}-1}{r-1}\,.$

Proof: Let $$S=\sum_{i=0}^{n} ar^i$$. Then, \begin{align*} rS &= r\sum_{i=0}^{n} ar^i \\ &= \sum_{i=0}^{n} ar^{i+1} \\ &= \sum_{j=1}^{n+1} ar^{j} \\ &= \left[\sum_{j=0}^{n} ar^{j}\right] + ar^{n+1} - a \\ &= S + ar^{n+1} - a \,. \end{align*} Now we can simplify the equation to \begin{align*} rS &= S + ar^{n+1} - a \\ (r-1)S &= a(r^{n+1} - 1) \\ S &= a\frac{r^{n+1} - 1}{r-1}\,.\quad ∎ \end{align*}

• We will have more use for summation as the course goes on… let's get those details as we need them.